#include <assert.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <gmp.h>
#include "pbc_fops.h"
#include "pbc_symtab.h"
#include "pbc_darray.h"
#include "pbc_field.h"
#include "pbc_poly.h"
#include "pbc_fp.h"
#include "pbc_fieldquadratic.h"
#include "pbc_hilbert.h"
#include "pbc_mnt.h"
#include "pbc_curve.h"
#include "pbc_pairing.h"
#include "pbc_g_param.h"
#include "pbc_param.h"
#include "pbc_tracker.h"
#include "pbc_memory.h"
#include "pbc_utils.h"

struct mnt_pairing_data_s {
    field_t Fq, Fqx, Fqd, Fqk;
    field_t Eq, Etwist;
    element_t nqrinv, nqrinv2;
    mpz_t tateexp;
    element_t xpowq, xpowq2, xpowq3, xpowq4;
};
typedef struct mnt_pairing_data_s mnt_pairing_data_t[1];
typedef struct mnt_pairing_data_s *mnt_pairing_data_ptr;

void g_param_init(g_param_ptr param)
{
    mpz_init(param->q);
    mpz_init(param->n);
    mpz_init(param->h);
    mpz_init(param->r);
    mpz_init(param->a);
    mpz_init(param->b);
    mpz_init(param->nk);
    mpz_init(param->hk);
    param->coeff = NULL;
    mpz_init(param->nqr);
}

void g_param_clear(g_param_ptr param)
{
    int i;
    mpz_clear(param->q);
    mpz_clear(param->n);
    mpz_clear(param->h);
    mpz_clear(param->r);
    mpz_clear(param->a);
    mpz_clear(param->b);
    mpz_clear(param->nk);
    mpz_clear(param->hk);
    mpz_clear(param->nqr);
    for (i=0; i<5; i++) {
	mpz_clear(param->coeff[i]);
    }
    pbc_free(param->coeff);
}

void g_param_out_str(FILE *stream, g_param_ptr p)
{
    int i;
    char s[80];
    param_out_type(stream, "g");
    param_out_mpz(stream, "q", p->q);
    param_out_mpz(stream, "n", p->n);
    param_out_mpz(stream, "h", p->h);
    param_out_mpz(stream, "r", p->r);
    param_out_mpz(stream, "a", p->a);
    param_out_mpz(stream, "b", p->b);
    param_out_mpz(stream, "nk", p->nk);
    param_out_mpz(stream, "hk", p->hk);
    for (i=0; i<5; i++) {
	sprintf(s, "coeff%d", i);
	param_out_mpz(stream, s, p->coeff[i]);
    }
    param_out_mpz(stream, "nqr", p->nqr);
}

void g_param_inp_generic (g_param_ptr p, fetch_ops_t fops, void *ctx)
{
    assert (fops);
    assert (ctx);
    symtab_t tab;
    char s[80];
    int i;

    symtab_init(tab);
    param_read_generic (tab, fops, ctx);

    lookup_mpz(p->q, tab, "q");
    lookup_mpz(p->n, tab, "n");
    lookup_mpz(p->h, tab, "h");
    lookup_mpz(p->r, tab, "r");
    lookup_mpz(p->a, tab, "a");
    lookup_mpz(p->b, tab, "b");
    lookup_mpz(p->nk, tab, "nk");
    lookup_mpz(p->hk, tab, "hk");
    lookup_mpz(p->nqr, tab, "nqr");

    p->coeff = pbc_realloc(p->coeff, sizeof(mpz_t) * 5);
    for (i=0; i<5; i++) {
	sprintf(s, "coeff%d", i);
	mpz_init(p->coeff[i]);
	lookup_mpz(p->coeff[i], tab, s);
    }

    param_clear_tab(tab);
    symtab_clear(tab);
}

static inline void d_miller_evalfn(element_t e0,
	element_t a, element_t b, element_t c,
	element_t Qx, element_t Qy)
{
    //a, b, c are in Fq
    //point Q is (Qx, Qy * sqrt(nqr)) where nqr is used to construct
    //the quadratic field extension Fqk of Fqd
    element_ptr re_out = fi_re(e0);
    element_ptr im_out = fi_im(e0);

    int i;
    int d = polymod_field_degree(re_out->field);
    for (i=0; i<d; i++) {
	element_mul(polymod_coeff(re_out, i), polymod_coeff(Qx, i), a);
	element_mul(polymod_coeff(im_out, i), polymod_coeff(Qy, i), b);
    }
    element_add(polymod_coeff(re_out, 0), polymod_coeff(re_out, 0), c);
}

static void cc_miller_no_denom_proj(element_t res, mpz_t q, element_t P,
	element_ptr Qx, element_ptr Qy)
{
    int m;
    element_t v;
    element_t Z;
    element_t a, b, c;
    element_t t0, t1;
    element_ptr t2 = a, t3 = b, t4 = c;
    element_t e0;
    element_t z, z2;
    element_ptr Zx, Zy;
    const element_ptr curve_a = curve_a_coeff(P);
    const element_ptr Px = curve_x_coord(P);
    const element_ptr Py = curve_y_coord(P);

    void proj_double(void)
    {
	//t0 = 3x^2 + (curve_a) z^4
	element_square(t0, Zx);
	//element_mul_si(t0, t0, 3);
	element_double(t1, t0);
	element_add(t0, t0, t1);
	element_square(t1, z2);
	element_mul(t1, t1, curve_a);
	element_add(t0, t0, t1);

	//z_out = 2 y z
	element_mul(z, Zy, z);
	//element_mul_si(z, z, 2);
	element_double(z, z);
	element_square(z2, z);

	//t1 = 4 x y^2
	element_square(t2, Zy);
	element_mul(t1, Zx, t2);
	//element_mul_si(t1, t1, 4);
	element_double(t1, t1);
	element_double(t1, t1);

	//x_out = t0^2 - 2 t1
	//element_mul_si(t3, t1, 2);
	element_double(t3, t1);
	element_square(Zx, t0);
	element_sub(Zx, Zx, t3);

	//t2 = 8y^4
	element_square(t2, t2);
	//element_mul_si(t2, t2, 8);
	element_double(t2, t2);
	element_double(t2, t2);
	element_double(t2, t2);

	//y_out = t0(t1 - x_out) - t2
	element_sub(t1, t1, Zx);
	element_mul(t0, t0, t1);
	element_sub(Zy, t0, t2);
    }

    void proj_mixin(void)
    {
	//t2 = Px z^2
	element_mul(t2, z2, Px);

	//t3 = Zx - t2
	element_sub(t3, Zx, t2);

	//t0 = Py z^3
	element_mul(t0, z2, Py);
	element_mul(t0, t0, z);

	//t1 = Zy - t0
	element_sub(t1, Zy, t0);

	//e7 = Zx + t2, use t2 to double for e7
	element_add(t2, Zx, t2);

	//e8 = Zy + t0, use t0 to double for e8
	element_add(t0, Zy, t0);

	//z = z t3
	element_mul(z, z, t3);
	element_square(z2, z);

	//Zx = t1^2 - e7 t3^2
	//t3 now holds t3^3,
	//t4 holds e7 t3^2
	element_square(t4, t3);
	element_mul(t3, t4, t3);
	element_square(Zx, t1);
	element_mul(t4, t2, t4);
	element_sub(Zx, Zx, t4);

	//t4 = e7 t3^2 - 2 Zx
	element_sub(t4, t4, Zx);
	element_sub(t4, t4, Zx);

	//Zy = (t4 t1 - e8 t3^3)/2
	element_mul(t4, t4, t1);
	element_mul(t0, t0, t3);
	element_sub(t4, t4, t0);
	element_halve(Zy, t4);
    }

    void do_tangent(void)
    {
	//a = -(3x^2 + cca z^4)
	//b = 2 y z^3
	//c = -(2 y^2 + x a)
	//a = z^2 a
	element_square(a, z2);
	element_mul(a, a, curve_a);
	element_square(b, Zx);
	//element_mul_si(b, b, 3);
	element_double(t0, b);
	element_add(b, b, t0);
	element_add(a, a, b);
	element_neg(a, a);

	element_mul(b, z, z2);
	element_mul(b, b, Zy);
	element_mul_si(b, b, 2);

	element_mul(c, Zx, a);
	element_mul(a, a, z2);
	element_square(t0, Zy);
	element_mul_si(t0, t0, 2);
	element_add(c, c, t0);
	element_neg(c, c);

	d_miller_evalfn(e0, a, b, c, Qx, Qy);
	element_mul(v, v, e0);
    }

    void do_line(void)
    {
	//a = -(Py z^3 - Zy)
	//b = Px z^3 - Zx z
	//c = Zx z Py - Zy Px;

	element_mul(t0, Zx, z);
	element_mul(t1, z2, z);

	element_mul(a, Py, t1);
	element_sub(a, Zy, a);

	element_mul(b, Px, t1);
	element_sub(b, b, t0);

	element_mul(t0, t0, Py);
	element_mul(c, Zy, Px);
	element_sub(c, t0, c);

	d_miller_evalfn(e0, a, b, c, Qx, Qy);
	element_mul(v, v, e0);
    }

    element_init(a, Px->field);
    element_init(b, a->field);
    element_init(c, a->field);
    element_init(t0, a->field);
    element_init(t1, a->field);
    element_init(e0, res->field);
    element_init(z, a->field);
    element_init(z2, a->field);
    element_set1(z);
    element_set1(z2);

    element_init(v, res->field);
    element_init(Z, P->field);

    element_set(Z, P);
    Zx = curve_x_coord(Z);
    Zy = curve_x_coord(Z);

    element_set1(v);
    m = mpz_sizeinbase(q, 2) - 2;

    for(;;) {
	do_tangent();
	if (!m) break;
	proj_double();
	if (mpz_tstbit(q, m)) {
	    do_line();
	    proj_mixin();
	}
	m--;
	element_square(v, v);
    }

    element_set(res, v);

    element_clear(v);
    element_clear(Z);
    element_clear(a);
    element_clear(b);
    element_clear(c);
    element_clear(t0);
    element_clear(t1);
    element_clear(e0);
    element_clear(z);
    element_clear(z2);
}

static void cc_miller_no_denom_affine(element_t res, mpz_t q, element_t P,
	element_ptr Qx, element_ptr Qy)
{
    int m;
    element_t v;
    element_t Z;
    element_t a, b, c;
    element_t t0;
    element_t e0;
    const element_ptr cca = curve_a_coeff(P);
    const element_ptr Px = curve_x_coord(P);
    const element_ptr Py = curve_y_coord(P);
    element_ptr Zx, Zy;

    /* TODO: when exactly is this not needed?
    void do_vertical(void)
    {
	mapbase(e0, Z->x);
	element_sub(e0, Qx, e0);
	element_mul(v, v, e0);
    }
    */

    void do_tangent(void)
    {
	//a = -(3 Zx^2 + cc->a)
	//b = 2 * Zy
	//c = -(2 Zy^2 + a Zx);

	element_square(a, Zx);
	element_mul_si(a, a, 3);
	element_add(a, a, cca);
	element_neg(a, a);

	element_add(b, Zy, Zy);

	element_mul(t0, b, Zy);
	element_mul(c, a, Zx);
	element_add(c, c, t0);
	element_neg(c, c);

	d_miller_evalfn(e0, a, b, c, Qx, Qy);
	element_mul(v, v, e0);
    }

    void do_line(void)
    {
	//a = -(B.y - A.y) / (B.x - A.x);
	//b = 1;
	//c = -(A.y + a * A.x);
	//but we'll multiply by B.x - A.x to avoid division

	element_sub(b, Px, Zx);
	element_sub(a, Zy, Py);
	element_mul(t0, b, Zy);
	element_mul(c, a, Zx);
	element_add(c, c, t0);
	element_neg(c, c);

	d_miller_evalfn(e0, a, b, c, Qx, Qy);
	element_mul(v, v, e0);
    }

    element_init(a, Px->field);
    element_init(b, a->field);
    element_init(c, a->field);
    element_init(t0, a->field);
    element_init(e0, res->field);

    element_init(v, res->field);
    element_init(Z, P->field);

    element_set(Z, P);
    Zx = curve_x_coord(Z);
    Zy = curve_y_coord(Z);

    element_set1(v);
    m = mpz_sizeinbase(q, 2) - 2;

    for(;;) {
	do_tangent();

	if (!m) break;

	element_double(Z, Z);
	if (mpz_tstbit(q, m)) {
	    do_line();
	    element_add(Z, Z, P);
	}
	m--;
	element_square(v, v);
    }

    element_set(res, v);

    element_clear(v);
    element_clear(Z);
    element_clear(a);
    element_clear(b);
    element_clear(c);
    element_clear(t0);
    element_clear(e0);
}

static void lucas_even(element_ptr out, element_ptr in, mpz_t cofactor)
//assumes cofactor is even
//mangles in
//in cannot be out
{
    element_t temp;
    element_init_same_as(temp, out);
    element_ptr in0 = fi_re(in);
    element_ptr in1 = fi_im(in);
    element_ptr v0 = fi_re(out);
    element_ptr v1 = fi_im(out);
    element_ptr t0 = fi_re(temp);
    element_ptr t1 = fi_im(temp);
    int j;

    element_set_si(t0, 2);
    element_double(t1, in0);

    element_set(v0, t0);
    element_set(v1, t1);

    j = mpz_sizeinbase(cofactor, 2) - 1;
    for (;;) {
	if (!j) {
	    element_mul(v1, v0, v1);
	    element_sub(v1, v1, t1);
	    element_square(v0, v0);
	    element_sub(v0, v0, t0);
	    break;
	}
	if (mpz_tstbit(cofactor, j)) {
	    element_mul(v0, v0, v1);
	    element_sub(v0, v0, t1);
	    element_square(v1, v1);
	    element_sub(v1, v1, t0);
	} else {
	    element_mul(v1, v0, v1);
	    element_sub(v1, v1, t1);
	    element_square(v0, v0);
	    element_sub(v0, v0, t0);
	}
	j--;
    }

    //assume cofactor = (q^2 - q + 1) / r is odd
    //thus v1 = V_k, v0 = V_{k-1}
    //     U = (P v1 - 2 v0) / (P^2 - 4)

    element_double(v0, v0);
    element_mul(in0, t1, v1);
    element_sub(in0, in0, v0);

    element_square(t1, t1);
    element_sub(t1, t1, t0);
    element_sub(t1, t1, t0);

    element_halve(v0, v1);
    element_div(v1, in0, t1);
    element_mul(v1, v1, in1);
}

static void cc_tatepower(element_ptr out, element_ptr in, pairing_t pairing)
{
    mnt_pairing_data_ptr p = pairing->data;
    element_t e0, e1, e2, e3;
    element_init(e0, p->Fqk);
    element_init(e1, p->Fqd);
    element_init(e2, p->Fqd);
    element_init(e3, p->Fqk);
    element_ptr e0re = fi_re(e0);
    element_ptr e0im = fi_im(e0);
    element_ptr e0re0 = ((element_t *) e0re->data)[0];
    element_ptr e0im0 = ((element_t *) e0im->data)[0];
    element_t *inre = fi_re(in)->data;
    element_t *inim = fi_im(in)->data;
    //see thesis
    void qpower(int sign) {
	polymod_const_mul(e2, inre[1], p->xpowq);
	element_set(e0re, e2);
	polymod_const_mul(e2, inre[2], p->xpowq2);
	element_add(e0re, e0re, e2);
	polymod_const_mul(e2, inre[3], p->xpowq3);
	element_add(e0re, e0re, e2);
	polymod_const_mul(e2, inre[4], p->xpowq4);
	element_add(e0re, e0re, e2);
	element_add(e0re0, e0re0, inre[0]);

	if (sign > 0) {
	    polymod_const_mul(e2, inim[1], p->xpowq);
	    element_set(e0im, e2);
	    polymod_const_mul(e2, inim[2], p->xpowq2);
	    element_add(e0im, e0im, e2);
	    polymod_const_mul(e2, inim[3], p->xpowq3);
	    element_add(e0im, e0im, e2);
	    polymod_const_mul(e2, inim[4], p->xpowq4);
	    element_add(e0im, e0im, e2);
	    element_add(e0im0, e0im0, inim[0]);
	} else {
	    polymod_const_mul(e2, inim[1], p->xpowq);
	    element_neg(e0im, e2);
	    polymod_const_mul(e2, inim[2], p->xpowq2);
	    element_sub(e0im, e0im, e2);
	    polymod_const_mul(e2, inim[3], p->xpowq3);
	    element_sub(e0im, e0im, e2);
	    polymod_const_mul(e2, inim[4], p->xpowq4);
	    element_sub(e0im, e0im, e2);
	    element_sub(e0im0, e0im0, inim[0]);
	}
    }
    qpower(1);
    element_set(e3, e0);
    element_set(e0re, fi_re(in));
    element_neg(e0im, fi_im(in));
    element_mul(e3, e3, e0);
    qpower(-1);
    element_mul(e0, e0, in);
    element_invert(e0, e0);
    element_mul(in, e3, e0);

    element_set(e0, in);
    lucas_even(out, e0, pairing->phikonr);

    element_clear(e0);
    element_clear(e1);
    element_clear(e2);
    element_clear(e3);
}

static void (*cc_miller_no_denom_fn)(element_t res, mpz_t q, element_t P,
	element_ptr Qx, element_ptr Qy);

static void cc_pairing(element_ptr out, element_ptr in1, element_ptr in2,
	pairing_t pairing)
{
    element_ptr Qbase = in2;
    element_t Qx, Qy;
    mnt_pairing_data_ptr p = pairing->data;

    element_init(Qx, p->Fqd);
    element_init(Qy, p->Fqd);
    //map from twist: (x, y) --> (v^-1 x, v^-(3/2) y)
    //where v is the quadratic nonresidue used to construct the twist
    element_mul(Qx, curve_x_coord(Qbase), p->nqrinv);
    //v^-3/2 = v^-2 * v^1/2
    element_mul(Qy, curve_y_coord(Qbase), p->nqrinv2);
    cc_miller_no_denom_fn(out, pairing->r, in1, Qx, Qy);
    cc_tatepower(out, out, pairing);
    element_clear(Qx);
    element_clear(Qy);
}

static int cc_is_almost_coddh(element_ptr a, element_ptr b,
	element_ptr c, element_ptr d,
	pairing_t pairing)
{
    int res = 0;
    element_t t0, t1, t2;
    element_t cx, cy;
    element_t dx, dy;
    mnt_pairing_data_ptr p = pairing->data;

    element_init(cx, p->Fqd);
    element_init(cy, p->Fqd);
    element_init(dx, p->Fqd);
    element_init(dy, p->Fqd);

    element_init(t0, pairing->GT);
    element_init(t1, pairing->GT);
    element_init(t2, pairing->GT);
    //map from twist: (x, y) --> (v^-1 x, v^-(3/2) y)
    //where v is the quadratic nonresidue used to construct the twist
    element_mul(cx, curve_x_coord(c), p->nqrinv);
    element_mul(dx, curve_x_coord(d), p->nqrinv);
    //v^-3/2 = v^-2 * v^1/2
    element_mul(cy, curve_y_coord(c), p->nqrinv2);
    element_mul(dy, curve_y_coord(d), p->nqrinv2);

    cc_miller_no_denom_fn(t0, pairing->r, a->data, dx, dy);
    cc_miller_no_denom_fn(t1, pairing->r, b->data, cx, cy);
    cc_tatepower(t0, t0, pairing);
    cc_tatepower(t1, t1, pairing);
    element_mul(t2, t0, t1);
    if (element_is1(t2)) {
	//g, g^x, h, h^-x case
	res = 1;
    } else {
	element_invert(t1, t1);
	element_mul(t2, t0, t1);
	if (element_is1(t2)) {
	    //g, g^x, h, h^x case
	    res = 1;
	}
    }
    element_clear(cx);
    element_clear(cy);
    element_clear(dx);
    element_clear(dy);
    element_clear(t0);
    element_clear(t1);
    element_clear(t2);
    return res;
}

struct pp_coeff_s {
    element_t a;
    element_t b;
    element_t c;
};
typedef struct pp_coeff_s pp_coeff_t[1];
typedef struct pp_coeff_s *pp_coeff_ptr;

static void g_pairing_pp_init(pairing_pp_t p, element_ptr in1, pairing_t pairing)
{
    element_ptr P = in1;
    const element_ptr Px = curve_x_coord(P);
    const element_ptr Py = curve_y_coord(P);
    element_t Z;
    int m;
    mnt_pairing_data_ptr info = pairing->data;
    element_t t0;
    element_t a, b, c;
    field_ptr Fq = info->Fq;
    pp_coeff_t *coeff;
    mpz_ptr q = pairing->r;
    pp_coeff_ptr pp;
    const element_ptr cca = curve_a_coeff(P);
    element_ptr Zx;
    element_ptr Zy;

    void store_abc(void)
    {
	element_init(pp->a, Fq);
	element_init(pp->b, Fq);
	element_init(pp->c, Fq);
	element_set(pp->a, a);
	element_set(pp->b, b);
	element_set(pp->c, c);
	pp++;
    }

    void do_tangent(void)
    {
	//a = -slope_tangent(Z.x, Z.y);
	//b = 1;
	//c = -(Z.y + a * Z.x);
	//but we multiply by 2*Z.y to avoid division

	//a = -Zx * (3 Zx + twicea_2) - a_4;
	//Common curves: a2 = 0 (and cc->a is a_4), so
	//a = -(3 Zx^2 + cc->a)
	//b = 2 * Zy
	//c = -(2 Zy^2 + a Zx);

	element_square(a, Zx);
	element_double(t0, a);
	element_add(a, a, t0);
	element_add(a, a, cca);
	element_neg(a, a);

	element_add(b, Zy, Zy);

	element_mul(t0, b, Zy);
	element_mul(c, a, Zx);
	element_add(c, c, t0);
	element_neg(c, c);

	store_abc();
    }

    void do_line(void)
    {
	//a = -(B.y - A.y) / (B.x - A.x);
	//b = 1;
	//c = -(A.y + a * A.x);
	//but we'll multiply by B.x - A.x to avoid division

	element_sub(b, Px, Zx);
	element_sub(a, Zy, Py);
	element_mul(t0, b, Zy);
	element_mul(c, a, Zx);
	element_add(c, c, t0);
	element_neg(c, c);

	store_abc();
    }

    element_init(Z, P->field);
    element_set(Z, P);
    Zx = curve_x_coord(Z);
    Zy = curve_y_coord(Z);

    element_init(t0, Fq);
    element_init(a, Fq);
    element_init(b, Fq);
    element_init(c, Fq);

    m = mpz_sizeinbase(q, 2) - 2;
    p->data = pbc_malloc(sizeof(pp_coeff_t) * 2 * m);
    coeff = (pp_coeff_t *) p->data;
    pp = coeff[0];

    for(;;) {
	do_tangent();

	if (!m) break;

	element_double(Z, Z);
	if (mpz_tstbit(q, m)) {
	    do_line();
	    element_add(Z, Z, P);
	}
	m--;
    }

    element_clear(t0);
    element_clear(a);
    element_clear(b);
    element_clear(c);
    element_clear(Z);
}

static void g_pairing_pp_clear(pairing_pp_t p)
{
    //TODO: better to store a sentinel value in p->data?
    mpz_ptr q = p->pairing->r;
    int m = mpz_sizeinbase(q, 2) + mpz_popcount(q) - 3;
    int i;
    pp_coeff_t *coeff = (pp_coeff_t *) p->data;
    pp_coeff_ptr pp;
    for (i=0; i<m; i++) {
	pp = coeff[i];
	element_clear(pp->a);
	element_clear(pp->b);
	element_clear(pp->c);
    }
    pbc_free(p->data);
}

static void g_pairing_pp_apply(element_ptr out, element_ptr in2, pairing_pp_t p)
{
    mpz_ptr q = p->pairing->r;
    mnt_pairing_data_ptr info = p->pairing->data;
    int m = mpz_sizeinbase(q, 2) - 2;
    pp_coeff_t *coeff = (pp_coeff_t *) p->data;
    pp_coeff_ptr pp = coeff[0];
    element_ptr Qbase = in2;
    element_t e0;
    element_t Qx, Qy;
    element_t v;
    element_init_GT(e0, p->pairing);
    element_init_GT(v, p->pairing);
    element_init(Qx, info->Fqd);
    element_init(Qy, info->Fqd);

    //map from twist: (x, y) --> (v^-1 x, v^-(3/2) y)
    //where v is the quadratic nonresidue used to construct the twist
    element_mul(Qx, curve_x_coord(Qbase), info->nqrinv);
    //v^-3/2 = v^-2 * v^1/2
    element_mul(Qy, curve_y_coord(Qbase), info->nqrinv2);

    element_set1(out);
    for(;;) {
	d_miller_evalfn(e0, pp->a, pp->b, pp->c, Qx, Qy);
	element_mul(out, out, e0);
	pp++;

	if (!m) break;

	if (mpz_tstbit(q, m)) {
	    d_miller_evalfn(e0, pp->a, pp->b, pp->c, Qx, Qy);
	    element_mul(out, out, e0);
	    pp++;
	}
	m--;
	element_square(out, out);
    }
    cc_tatepower(out, out, p->pairing);

    element_clear(e0);
    element_clear(Qx);
    element_clear(Qy);
    element_clear(v);
}

static void g_pairing_ellnet(element_ptr out, element_ptr in1, element_ptr in2,
	pairing_t pairing)
//in1, in2 are from E(F_q), out from F_q^2
//uses elliptic nets (see Stange)
{
    mnt_pairing_data_ptr p = pairing->data;

    const element_ptr a = curve_a_coeff(in1);
    const element_ptr b = curve_b_coeff(in1);

    element_ptr x = curve_x_coord(in1);
    element_ptr y = curve_y_coord(in1);

    element_ptr x2 = curve_x_coord(in2);
    element_ptr y2 = curve_y_coord(in2);

    //we map (x2,y2) to (-x2, i y2) before pairing
    //notation: cmi means c_{k-i}, ci means c_{k+i}
    element_t cm3, cm2, cm1, c0, c1, c2, c3, c4;
    element_t dm1, d0, d1;
    element_t A, B, C;

    element_init_same_as(cm3, x);
    element_init_same_as(cm2, x);
    element_init_same_as(cm1, x);
    element_init_same_as(c0, x);
    element_init_same_as(c1, x);
    element_init_same_as(c2, x);
    element_init_same_as(c3, x);
    element_init_same_as(c4, x);
    element_init_same_as(C, x);

    element_init_same_as(dm1, out);
    element_init_same_as(d0, out);
    element_init_same_as(d1, out);
    element_init_same_as(A, out);
    element_init_same_as(B, out);

    // c1 = 2y
    // cm3 = -2y
    element_double(c1, y);
    element_neg(cm3, c1);

    //use c0, cm1, cm2, C, c4 as temp variables for now
    //compute c3, c2
    element_square(cm2, x);
    element_square(C, cm2);
    element_mul(cm1, b, x);
    element_double(cm1, cm1);
    element_square(c4, a);

    element_mul(c2, cm1, cm2);
    element_double(c2, c2);
    element_mul(c0, a, C);
    element_add(c2, c2, c0);
    element_mul(c0, c4, cm2);
    element_sub(c2, c2, c0);
    element_double(c0, c2);
    element_double(c0, c0);
    element_add(c2, c2, c0);

    element_mul(c0, cm1, a);
    element_square(c3, b);
    element_double(c3, c3);
    element_double(c3, c3);
    element_add(c0, c0, c3);
    element_double(c0, c0);
    element_mul(c3, a, c4);
    element_add(c0, c0, c3);
    element_sub(c2, c2, c0);
    element_mul(c0, cm2, C);
    element_add(c3, c0, c2);
    element_mul(c3, c3, c1);
    element_double(c3, c3);

    element_mul(c0, a, cm2);
    element_add(c0, c0, cm1);
    element_double(c0, c0);
    element_add(c0, c0, C);
    element_double(c2, c0);
    element_add(c0, c0, c2);
    element_sub(c2, c0, c4);

    // c0 = 1
    // cm2 = -1
    element_set1(c0);
    element_neg(cm2, c0);

    // c4 = c_5 = c_2^3 c_4 - c_3^3 = c1^3 c3 - c2^3
    element_square(C, c1);
    element_mul(c4, C, c1);
    element_mul(c4, c4, c3);
    element_square(C, c2);
    element_mul(C, C, c2);
    element_sub(c4, c4, C);

    //compute A, B, d1

    element_mul(fi_re(d0), x2, p->nqrinv);
    element_neg(A, d0);
    element_add(polymod_coeff(fi_re(A), 0), polymod_coeff(fi_re(A), 0), x);

    element_double(C, x);
    element_add(polymod_coeff(fi_re(d0), 0), polymod_coeff(fi_re(d0), 0), C);

    element_square(dm1, A);
    element_mul(dm1, d0, dm1);

    element_mul(fi_im(d1), y2, p->nqrinv2);
    element_set(polymod_coeff(fi_re(d1), 0), y);

    element_square(d1, d1);
    element_sub(d1, dm1, d1);
    element_invert(B, d1);

    element_invert(A, A);

    element_mul(fi_im(d1), y2, p->nqrinv2);
    element_set0(fi_re(d1));
    element_neg(polymod_coeff(fi_re(d1), 0), y);
    element_mul(d1, d1, A);
    element_square(d1, d1);
    element_sub(d1, d0, d1);

    // cm1 = 0
    // C = (2y)^-1
    element_set0(cm1);
    element_invert(C, c1);

    element_set1(dm1);
    element_set1(d0);

    element_t sm2, sm1;
    element_t s0, s1, s2, s3;
    element_t tm2, tm1;
    element_t t0, t1, t2, t3;
    element_t e0, e1;
    element_t u, v;

    element_init_same_as(sm2, x);
    element_init_same_as(sm1, x);
    element_init_same_as(s0, x);
    element_init_same_as(s1, x);
    element_init_same_as(s2, x);
    element_init_same_as(s3, x);

    element_init_same_as(tm2, x);
    element_init_same_as(tm1, x);
    element_init_same_as(t0, x);
    element_init_same_as(t1, x);
    element_init_same_as(t2, x);
    element_init_same_as(t3, x);

    element_init_same_as(e0, x);
    element_init_same_as(e1, x);

    element_init_same_as(u, d0);
    element_init_same_as(v, d0);

    int m = mpz_sizeinbase(pairing->r, 2) - 2;
    for (;;) {
	element_square(sm2, cm2);
	element_square(sm1, cm1);
	element_square(s0, c0);
	element_square(s1, c1);
	element_square(s2, c2);
	element_square(s3, c3);

	element_mul(tm2, cm3, cm1);
	element_mul(tm1, cm2, c0);
	element_mul(t0, cm1, c1);
	element_mul(t1, c0, c2);
	element_mul(t2, c1, c3);
	element_mul(t3, c2, c4);

	element_square(u, d0);
	element_mul(v, dm1, d1);

	if (mpz_tstbit(pairing->r, m)) {
	    //double-and-add
	    element_mul(e0, t0, sm2);
	    element_mul(e1, tm2, s0);
	    element_sub(cm3, e0, e1);
	    element_mul(cm3, cm3, C);

	    element_mul(e0, t0, sm1);
	    element_mul(e1, tm1, s0);
	    element_sub(cm2, e0, e1);

	    element_mul(e0, t1, sm1);
	    element_mul(e1, tm1, s1);
	    element_sub(cm1, e0, e1);
	    element_mul(cm1, cm1, C);

	    element_mul(e0, t1, s0);
	    element_mul(e1, t0, s1);
	    element_sub(c0, e0, e1);

	    element_mul(e0, t2, s0);
	    element_mul(e1, t0, s2);
	    element_sub(c1, e0, e1);
	    element_mul(c1, c1, C);

	    element_mul(e0, t2, s1);
	    element_mul(e1, t1, s2);
	    element_sub(c2, e0, e1);

	    element_mul(e0, t3, s1);
	    element_mul(e1, t1, s3);
	    element_sub(c3, e0, e1);
	    element_mul(c3, c3, C);

	    element_mul(e0, t3, s2);
	    element_mul(e1, t2, s3);
	    element_sub(c4, e0, e1);

	    polymod_const_mul(fi_re(out), t0,  fi_re(u));
	    polymod_const_mul(fi_im(out), t0,  fi_im(u));
	    polymod_const_mul(fi_re(dm1), s0,  fi_re(v));
	    polymod_const_mul(fi_im(dm1), s0,  fi_im(v));
	    element_sub(dm1, dm1, out);

	    polymod_const_mul(fi_re(out), t1, fi_re(u));
	    polymod_const_mul(fi_im(out), t1, fi_im(u));
	    polymod_const_mul(fi_re(d0), s1, fi_re(v));
	    polymod_const_mul(fi_im(d0), s1, fi_im(v));
	    element_sub(d0, d0, out);
	    element_mul(d0, d0, A);

	    polymod_const_mul(fi_re(out), t2, fi_re(u));
	    polymod_const_mul(fi_im(out), t2, fi_im(u));
	    polymod_const_mul(fi_re(d1), s2, fi_re(v));
	    polymod_const_mul(fi_im(d1), s2, fi_im(v));
	    element_sub(d1, d1, out);
	    element_mul(d1, d1, B);
	} else {
	    //double
	    element_mul(e0, tm1, sm2);
	    element_mul(e1, tm2, sm1);
	    element_sub(cm3, e0, e1);

	    element_mul(e0, t0, sm2);
	    element_mul(e1, tm2, s0);
	    element_sub(cm2, e0, e1);
	    element_mul(cm2, cm2, C);

	    element_mul(e0, t0, sm1);
	    element_mul(e1, tm1, s0);
	    element_sub(cm1, e0, e1);

	    element_mul(e0, t1, sm1);
	    element_mul(e1, tm1, s1);
	    element_sub(c0, e0, e1);
	    element_mul(c0, c0, C);

	    element_mul(e0, t1, s0);
	    element_mul(e1, t0, s1);
	    element_sub(c1, e0, e1);

	    element_mul(e0, t2, s0);
	    element_mul(e1, t0, s2);
	    element_sub(c2, e0, e1);
	    element_mul(c2, c2, C);

	    element_mul(e0, t2, s1);
	    element_mul(e1, t1, s2);
	    element_sub(c3, e0, e1);

	    element_mul(e0, t3, s1);
	    element_mul(e1, t1, s3);
	    element_sub(c4, e0, e1);
	    element_mul(c4, c4, C);

	    polymod_const_mul(fi_re(out), tm1, fi_re(u));
	    polymod_const_mul(fi_im(out), tm1, fi_im(u));
	    polymod_const_mul(fi_re(dm1), sm1, fi_re(v));
	    polymod_const_mul(fi_im(dm1), sm1, fi_im(v));
	    element_sub(dm1, dm1, out);

	    polymod_const_mul(fi_re(out), t0, fi_re(u));
	    polymod_const_mul(fi_im(out), t0, fi_im(u));
	    polymod_const_mul(fi_re(d0), s0, fi_re(v));
	    polymod_const_mul(fi_im(d0), s0, fi_im(v));
	    element_sub(d0, d0, out);

	    polymod_const_mul(fi_re(out), t1, fi_re(u));
	    polymod_const_mul(fi_im(out), t1, fi_im(u));
	    polymod_const_mul(fi_re(d1), s1, fi_re(v));
	    polymod_const_mul(fi_im(d1), s1, fi_im(v));
	    element_sub(d1, d1, out);
	    element_mul(d1, d1, A);
	}
	if (!m) break;
	m--;
    }
    // since c_k lies base field
    // it gets killed by the final powering
    //element_invert(c1, c1);
    //element_mul(fi_re(d1), fi_re(d1), c1);
    //element_mul(fi_im(d1), fi_im(d1), c1);

    cc_tatepower(out, d1, pairing);

    element_clear(dm1);
    element_clear(d0);
    element_clear(d1);

    element_clear(cm3);
    element_clear(cm2);
    element_clear(cm1);
    element_clear(c0);
    element_clear(c1);
    element_clear(c2);
    element_clear(c3);
    element_clear(c4);

    element_clear(sm2);
    element_clear(sm1);
    element_clear(s0);
    element_clear(s1);
    element_clear(s2);
    element_clear(s3);

    element_clear(tm2);
    element_clear(tm1);
    element_clear(t0);
    element_clear(t1);
    element_clear(t2);
    element_clear(t3);

    element_clear(e0);
    element_clear(e1);
    element_clear(A);
    element_clear(B);
    element_clear(C);
    element_clear(u);
    element_clear(v);
}

void g_pairing_clear(pairing_t pairing)
{
    mnt_pairing_data_ptr p = pairing->data;

    element_clear(p->xpowq);
    element_clear(p->xpowq2);
    element_clear(p->xpowq3);
    element_clear(p->xpowq4);
    mpz_clear(p->tateexp);
    mpz_clear(pairing->phikonr);

    field_clear(p->Etwist);
    field_clear(p->Eq);
    element_clear(p->nqrinv);
    element_clear(p->nqrinv2);
    field_clear(p->Fqk);
    field_clear(p->Fqd);
    field_clear(p->Fqx);
    field_clear(p->Fq);
    field_clear(pairing->Zr);
    mpz_clear(pairing->r);
    pbc_free(p);
}

static void g_pairing_option_set(pairing_t pairing, char *key, char *value)
{
    UNUSED_VAR(pairing);
    if (!strcmp(key, "method")) {
	if (!strcmp(value, "miller")) {
	    cc_miller_no_denom_fn = cc_miller_no_denom_proj;
	} else if (!strcmp(value, "miller-affine")) {
	    cc_miller_no_denom_fn = cc_miller_no_denom_affine;
	} else if (!strcmp(value, "shipsey-stange")) {
	    pairing->map = g_pairing_ellnet;
	}
    }
}

void pairing_init_g_param(pairing_t pairing, g_param_t param)
{
    mnt_pairing_data_ptr p;
    element_t a, b;
    element_t irred;
    int i;

    mpz_init(pairing->r);
    mpz_set(pairing->r, param->r);
    field_init_fp(pairing->Zr, pairing->r);
    pairing->map = cc_pairing;
    pairing->is_almost_coddh = cc_is_almost_coddh;

    p =	pairing->data = pbc_malloc(sizeof(mnt_pairing_data_t));
    field_init_fp(p->Fq, param->q);
    element_init(a, p->Fq);
    element_init(b, p->Fq);
    element_set_mpz(a, param->a);
    element_set_mpz(b, param->b);
    field_init_curve_ab(p->Eq, a, b, pairing->r, param->h);

    field_init_poly(p->Fqx, p->Fq);
    element_init(irred, p->Fqx);
    poly_alloc(irred, 5 + 1);
    for (i=0; i<5; i++) {
	element_set_mpz(poly_coeff(irred, i), param->coeff[i]);
    }
    element_set1(poly_coeff(irred, 5));

    field_init_polymod(p->Fqd, irred);
    element_clear(irred);

    p->Fqd->nqr = pbc_malloc(sizeof(element_t));
    element_init(p->Fqd->nqr, p->Fqd);
    element_set_mpz(((element_t *) p->Fqd->nqr->data)[0], param->nqr);

    field_init_quadratic(p->Fqk, p->Fqd);

    { //compute phi(k)/r = (q^4 - q^3 + ... + 1)/r 
	element_ptr e = p->xpowq;
	mpz_t z0;
	mpz_ptr q = param->q;
	mpz_ptr z = pairing->phikonr;
	mpz_init(z);
	mpz_init(z0);
	mpz_set_ui(z, 1);
	mpz_sub(z, z, q);
	mpz_mul(z0, q, q);
	mpz_add(z, z, z0);
	mpz_mul(z0, z0, q);
	mpz_sub(z, z, z0);
	mpz_mul(z0, z0, q);
	mpz_add(z, z, z0);
	mpz_clear(z0);
	mpz_divexact(z, z, pairing->r);

	element_init(e, p->Fqd);
	element_init(p->xpowq2, p->Fqd);
	element_init(p->xpowq3, p->Fqd);
	element_init(p->xpowq4, p->Fqd);
	element_set1(((element_t *) e->data)[1]);
	element_pow_mpz(e, e, q);

	element_square(p->xpowq2, p->xpowq);
	element_square(p->xpowq4, p->xpowq2);
	element_mul(p->xpowq3, p->xpowq2, p->xpowq);
    }
    mpz_init(p->tateexp);
    mpz_sub_ui(p->tateexp, p->Fqk->order, 1);
    mpz_divexact(p->tateexp, p->tateexp, pairing->r);

    field_init_curve_ab_map(p->Etwist, p->Eq, element_field_to_polymod, p->Fqd, pairing->r, NULL);

    twist_curve(p->Etwist);

    element_init(p->nqrinv, p->Fqd);
    element_invert(p->nqrinv, field_get_nqr(p->Fqd));
    element_init(p->nqrinv2, p->Fqd);
    element_square(p->nqrinv2, p->nqrinv);

    pairing->G1 = p->Eq;
    pairing->G2 = p->Etwist;

    pairing->GT = p->Fqk;

    cc_miller_no_denom_fn = cc_miller_no_denom_affine;
    pairing->option_set = g_pairing_option_set;
    pairing->pp_init = g_pairing_pp_init;
    pairing->pp_clear = g_pairing_pp_clear;
    pairing->pp_apply = g_pairing_pp_apply;

    pairing->clear_func = g_pairing_clear;

    element_clear(a);
    element_clear(b);
}

static void compute_cm_curve(g_param_ptr param, cm_info_ptr cm)
    //computes a curve and sets fp to the field it is defined over
    //using the complex multiplication method, where cm holds
    //the appropriate information (e.g. discriminant, field order)
{
    darray_t coefflist;
    element_t hp, root;
    field_t fp, fpx;
    int i, n;
    field_t cc;

    field_init_fp(fp, cm->q);
    field_init_poly(fpx, fp);
    element_init(hp, fpx);

    darray_init(coefflist);

    hilbert_poly(coefflist, cm->D);

    n = coefflist->count;
    poly_alloc(hp, n);
    for (i=0; i<n; i++) {
	element_set_mpz(poly_coeff(hp, i), coefflist->item[i]);
    }

    hilbert_poly_clear(coefflist);

    darray_clear(coefflist);
    //TODO: remove x = 0, 1728 roots
    //TODO: what if there's no roots?
    //printf("hp ");
    //element_out_str(stdout, 0, hp);
    //printf("\n");

    element_init(root, fp);
    findroot(root, hp);
    //printf("root = ");
    //element_out_str(stdout, 0, root);
    //printf("\n");
    element_clear(hp);
    field_clear(fpx);

    //the root is the j-invariant of our desired curve
    field_init_curve_j(cc, root, cm->n, NULL);
    element_clear(root);

    //we may need to twist it however
    {
	element_t P;

	//pick a random point P and see if it has the right order
	element_init(P, cc);
	element_random(P);
	element_mul_mpz(P, P, cm->n);
	//element_printf("P = %B", P);
	//if not, we twist the curve
	if (!element_is0(P)) {
	    twist_curve(cc);
	}
	element_clear(P);
    }

    mpz_set(param->q, cm->q);
    mpz_set(param->n, cm->n);
    mpz_set(param->h, cm->h);
    mpz_set(param->r, cm->r);
    element_to_mpz(param->a, curve_field_a_coeff(cc));
    element_to_mpz(param->b, curve_field_b_coeff(cc));
    {
	mpz_t z;
	mpz_init(z);
	//compute order of curve in F_q^k
	//n = q - t + 1 hence t = q - n + 1
	mpz_sub(z, param->q, param->n);
	mpz_add_ui(z, z, 1);
	compute_trace_n(z, param->q, z, 10);
	mpz_pow_ui(param->nk, param->q, 10);
	mpz_sub_ui(z, z, 1);
	mpz_sub(param->nk, param->nk, z);
	mpz_mul(z, param->r, param->r);
	mpz_divexact(param->hk, param->nk, z);
	mpz_clear(z);
    }
    field_clear(cc);
    field_clear(fp);
}

void g_param_from_cm(g_param_t param, cm_info_ptr cm)
{
    field_t Fq, Fqx, Fqd;
    element_t irred, nqr;
    int i;

    compute_cm_curve(param, cm);

    field_init_fp(Fq, param->q);
    field_init_poly(Fqx, Fq);
    element_init(irred, Fqx);
    do {
	poly_random_monic(irred, 5);
    } while (!poly_is_irred(irred));
    field_init_polymod(Fqd, irred);

    //find a quadratic nonresidue of Fqd lying in Fq
    element_init(nqr, Fqd);
    do {
	element_random(((element_t *) nqr->data)[0]);
    } while (element_is_sqr(nqr));

    param->coeff = pbc_realloc(param->coeff, sizeof(mpz_t) * 5);

    for (i=0; i<5; i++) {
	mpz_init(param->coeff[i]);
	element_to_mpz(param->coeff[i], poly_coeff(irred, i));
    }
    element_to_mpz(param->nqr, ((element_t *) nqr->data)[0]);

    element_clear(nqr);
    element_clear(irred);

    field_clear(Fqx);
    field_clear(Fqd);
    field_clear(Fq);
}
